Signs of the Sine Illusion
Why We Need to Care
Example
8-hour Average Ozone Levels in Houston, TX by temperature at Hobby Airport
Example
Residual Ozone Levels in Houston, TX by temperature at Hobby Airport
Explaining the Illusion
The sine illusion results from misapplication of a three-dimensional visual heuristic to ambiguous two-dimensional images 
Explaining the Illusion
The sine illusion results from misapplication of a three-dimensional visual heuristic to ambiguous two-dimensional images 
In this figure, the vanishing point has been moved towards infinity; the lines are straight and closer to the appearance of the sine illusion. The three-dimensional appearance is still intact.
Explaining the Illusion
Context Matters
- Cleveland and McGill (1984) demonstrated that we can compare the length of two lines accurately
- Day and Stecher (1991) showed that the illusion persists if there is curvature in the underlying function
Geometry of the Sine Illusion
Geometry of the Sine Illusion
We perceive the orthogonal width of the implied surface
The orthogonal width is a function of the x and y range as well as the aspect ratio of the plot.
The perceived orthogonal width is also a function of the slope of the line tangent to the underlying function curve.
Correcting the Illusion
- Trend Removal
Remove the underlying function, plotting the curve and the residuals separately
- X Axis Correction
Reparameterize the x-axis in terms of the slope, so that the absolute slope doesn’t change
- Y Axis Correction
Change the plotted line length (or spread) so that the perceived orthogonal width corresponds to the original (data) line length
Trend Removal
Trend Removal
Trend Removal
It is hard to re-create this graphic with separate curves that still provide all of the information
Let \(a\) and \(b\) be the minimum and maximum of the \(x\)-range under consideration.
For any value \(x \in (a,b)\) the following transformation results in a function with constant absolute slope:
\[
(f \circ T)(x) = a + (b-a)\left(\int_{a}^x |f^\prime(z)| dz\right)\left/\left(\int_{a}^{b}|f^\prime(z)| dz\right)\right.
\]
Shrinkage factor \(w \in (0,1)\): allows a less extreme approach to counteracting the illusion
\[(f \circ T_w)(x) = (1-w) \cdot x + w \cdot (f \circ T)(x)\]
- \(w=1\) is a full transformation
- smaller values of \(w\) indicate a less severe correction
- Under weaker transformations the data more closely reflect the original function \(f(x)\).
General Correction
If we extend the line length so that the extant width matches the original vertical length, our perceptions will match the original data.
General Correction
The function describing the orthogonal line through \((x_o, f(x_o))\) is given in point-vector form as
\[
{x_o \choose f(x_o)} + \lambda {f^\prime(x_o) \choose 1}
\]
for any real-valued \(\lambda\).
General Correction
Point vector form allows us to solve for \(\lambda\) easily, giving the extant (half) widths as:
\[
|\lambda| \sqrt{1 + f^\prime(x_o)^2}
\]
This equation describes the quantity that we perceive rather than the quantity that we want to display (\(\ell/2\))
General Correction
The general correction factor is thus
\[
\ell/2 \cdot \left(|\lambda| \sqrt{1 + f^\prime(x_o)^2}\right)^{-1}
\]
This yields two solutions; one for positive and one for negative values of \(\lambda\) corresponding to upper and lower (half) extant width.
General Correction
In order to get actual numeric values for \(\lambda\), we need to find end points \(f_1\) and \(f_2\). This system of equations provides solutions for those points:
\[
x - x_o = \lambda f^\prime(x_o)
\]
\[
f(x) - f(x_o) = -\lambda \pm \ell/2
\]
Solving these equations requires numerical optimization; we will use linear and quadratic taylor series to simplify the optimization processs.
Linear and Quadratic Approximations
Substituting the endpoints \((x_1, y_1)\) and \((x_2, y_2)\) into the general correction factor produces the linear and quadratic corrections to the sine illusion
Linear Correction
\[f(x)\approx f(x_0) + (x-x_0) f^\prime(x_0)\]
The correction factor is then
\[\ell_{new}(x_0) = \ell_{old}\sqrt{1+f^\prime(x_0)^2}\]
Linear Correction
Quadratic Correction
\[f(x) \approx f(x_0) + f^\prime(x_0)(x-x_0) + 1/2 f^{\prime\prime}(x_0)(x-x_0)^2\]
The general correction system of equation simplifies to
\[
f^{\prime\prime}(x_0) f^\prime(x_0)^2 \lambda^2 + 2(f^\prime(x_0)^2 + 1) \lambda \pm \ell = 0,
\]
Quadratic Correction
The corrections for the half lengths are then:
\[\begin{eqnarray}
\ell_{\text{new}_1}(x_0) &=& 1 /2 \cdot \left(v + \sqrt{ v^2 + f^{\prime\prime}(x_0) f^\prime(x_0)^2\cdot \ell_{\text{old}}}\right) \cdot v^{-1/2} \\
\ell_{\text{new}_2}(x_0) &=& 1 /2 \cdot \left(v + \sqrt{ v^2 - f^{\prime\prime}(x_0) f^\prime(x_0)^2\cdot \ell_{\text{old}}}\right) \cdot v^{-1/2}
\end{eqnarray}\]
where \(v = 1 + f^\prime(x_0)^2\)
In the quadratic correction, each half-length is corrected separately, producing a more robust correction
Quadratic Correction
The y-axis transformation can be weighted in the same manner as the x-axis transformation.
A Shiny applet was created to explore the x and y corrections.
Testing the Corrections
Goal : Determine the strength of the Sine Illusion by measuring how much correction is required for viewers to say that the lines are of equal length.
A different Shiny applet was created to allow users to manipulate the stimuli using fine-grained adjustments to the weight value.
Participants
- Using the shiny applet, users could manipulate the weight value presented using -/+ buttons until they were satisfied that the lines were of equal length. The trial was finished when users selected the ‘submit’ button.
Data Collection
User identification information: a ‘fingerprint’ consisting of hashed browser and computer characteristics was used to identify unique users
IP address localization (34.45.38.XX) provided location information
Every user interaction was recorded with a timestamp
Trial finished when user clicked either ‘submit’ or ‘skip’ to opt-out of the trial.
Experiment Design
- 12 (or more) trials, 6 of each correction type
- Each user completed trials starting at 0 and 1 for both correction types
- Additional trials were selected using starting weights between 0.25 and 0.75, with point density highest around 0.6
- After 12 trials, weights were chosen from starting weights between 0 and 1
Data Inclusion Criteria
Trial recorded at least two user interactions:
The user must adjust the weight value at least once and then click the submit button.
User completed at least 4 trials
User selected a weight value that was not severely over-corrected or under corrected (i.e. weight value selected was plausible)
Data Inclusion Criteria
Implausibly Over-corrected and Under-corrected Answers
Results
Included Data
Once exclusion criteria were applied, our data consisted of 125 participants who completed 1210 valid trials.
Psychophysics Model
Psychophysics Model
Let \(\gamma_X\) represent the optimal weight value for the \(X\)-correction
and \(\gamma_Y\) represent the optimal weight value for the \(Y\) correction.
\(\gamma_\ast = \frac{1}{2}(w_0 + w_1)\)
where \(w_0\) is the preferred weight when starting at 0, and \(w_1\) is the preferred weight when starting at 1.
Psychophysics Model
Random Effects Model
-
\(W_{ij}\) the weight chosen by participant \(i\) on trial \(j\)
\[1 \le i \le 125, 1 \le j \le n_i\]
-
\(T(i,j)\) the correction type, where \(T(i,j) \in \{X, Y\}\)
-
Starting weight \(X_{ij}\)
-
\(\alpha_\ast\), the lowest acceptable weight for correction type \(\ast\)
-
\(\beta\), the acceptable weight value interval width
-
Participant-level random intercept \(\gamma_{i, \ast}\)
Random Effects Model
\[
W_{ij} = \alpha_{T(i,j)} + \beta X_{ij} + \gamma_{i, T(i,j)} + \epsilon_{ij}\]
\[\gamma_{iX} \stackrel{\text{ i.i.d.}}{\sim} N(0, \eta_X^2) \ \ \ \ \ \ \ \ \gamma_{iY} \stackrel{\text{ i.i.d.}}{\sim} N(0, \eta_Y^2) \]
\[\epsilon_{ij} \stackrel{\text{ i.i.d.}}{\sim} N(0, \sigma^2) \ \ \ \ \ \ \ \ \text{Cov}(\gamma, \epsilon) = 0\]
The range of acceptable values is
\[(\alpha_\ast, \alpha_\ast + \beta)\]
Random Effects Model
\[
W_{ij} = \alpha_{T(i,j)} + \beta X_{ij} + \gamma_{i, T(i,j)} + \epsilon_{ij}\]
\[\gamma_{iX} \stackrel{\text{ i.i.d.}}{\sim} N(0, \eta_X^2) \ \ \ \ \ \ \ \ \gamma_{iY} \stackrel{\text{ i.i.d.}}{\sim} N(0, \eta_Y^2) \]
\[\epsilon_{ij} \stackrel{\text{ i.i.d.}}{\sim} N(0, \sigma^2) \ \ \ \ \ \ \ \ \text{Cov}(\gamma, \epsilon) = 0\]
We can compare this model to the psychophysics model using the midpoint of this interval, \[\alpha_\ast+\beta/2\]
Random Effects Model
Results
|
Transformation
|
Threshold
|
Estimate
|
95% C.I.
|
|
X
|
Lower
|
0.097
|
(0.048, 0.149)
|
| |
Upper
|
0.625
|
(0.570, 0.684)
|
|
Y
|
Lower
|
0.143
|
(0.094, 0.187)
|
| |
Upper
|
0.671
|
(0.622, 0.717)
|
Random Effects Model
Conclusions
Either correction is preferrable to an uncorrected graph
Corrections do not have to be fully applied to break the illusion’s power
The sine illusion is strong enough to make participants think that lines of unequal length are equal
Importance to Statistical Graphics
We can’t judge variability accurately when there is a nonlinear trend. Knowing is half the battle; having tools to screen for this effect could also be helpful.
